Hi there,
I've taken out the names of her children, as it seems some are living and also just to focus on Elizabeth's parentage.
It looks like Eric W Valentine's marriage was registered in Fayid, Egypt in 1951 (there is a matching Elizabeth Rogers on the free search)
https://www.findmypast.com.au/search/results?firstname=eric&firstname_variants=true&lastname=valentine&sourcecategory=life%20events%20(bmds)&keywordsplace=egypt&keywordsplace_proximity=5&sid=999https://www.gro.gov.uk/gro/content/certificates/app_select.asp You can try to get hold of her marriage certificate, this should tell you at least her father's name.
A birth July 11 1927, if in England or Wales, would be registered in Sep quarter 1927.
One option would be William G Rogers and Sarah F Foster who married in St Olave in 1918 They had four children including an Elizabeth born in the right quarter who was their last child.
From 1921 census (again, just free index) Sarah was older than William and would have
William G - born 1888 Woolwich
Sarah - born 1882 Rotherhithe
William G - born 1921 Rotherhithe
It appears Sarah died in 1938, aged 56. Ths would line up with the family stories. Any of these names/locations ringing a bell?
In 1939 register I think the family are at
6 Stalham Street, Bermondsey, Southwark (Elizabeth is not shown due to her birthyear but you might see a redacted line for her)
This reveals another sister, Mary C Rogers (later Short), born 1919, who married in 1942.